Phy5670/Bethe Ansatz testing

Bethe-Ansatz for 1D-Heisenberg Model (Ref[1])

The Bethe ansatz was originally developed for the one-dimensional Heisenberg model with nearest neighbor interaction and periodic boundary conditions:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H=-J\sum^N_{n=1}\vec{S}_n\cdot \vec{S}_{n+1}=-J\sum^N_{n=1}\left[\frac{1}{2}(S_n^+S^-_{n+1} + S_n^-S^+_{n+1})+S^z_nS^z_{n+1} \right] }

where J>0 for Ferromagnet and J<0 for Anti-Ferromagnet.

In the ferromagnetic ground state, all spins are aligned in one direction along. We take it as z-direction. Thus, the ground state can be described as:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |F\rangle = |\uparrow\uparrow\uparrow...\uparrow\rangle }

If now two of the up spins are flipped to down spins, and if these 2 spins are in position n1 and n2, we can specify the state as:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |n_1n_2\rangle = |\uparrow\uparrow\underbrace{\downarrow}_{n_1}\uparrow..\uparrow\underbrace{\downarrow}_{n_2} \uparrow...\uparrow\rangle }

As Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [S_{z},H]=0 } , so the eigenstates of the Hamiltonian are given by the superpositions of states which have same number of flipped spins with the spin being put at different combination of positions.

case of only one flipped spin

Eigenstate is given by superpositions of states with only one flipped spin at the lattice site n:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\Psi\rangle = \sum^N_{n=1}a(n)|n\rangle }

The eigenvectors are solutions of the stationary Schrödinger equation. By comparing coefficients, we obtain N difference equations for the coefficients a(n):

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2\left [E-E_{0}a(n)\right ]=J\left [2a(n)-a(n-1)-a(n+1)\right ] }

By applying periodic boundary condition Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a(n+N)=a(n)} we obtain plane wave solution:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a(n)=e^{ikn}, \qquad k=\frac{2\pi}{N}m \qquad \text{where} \quad m=0,1,... N-1 }

Thus, the eigenvectors are given by superpositions of states with only one flipped spin. The energy of these states follows from the Schrodinger equation:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E=E_{0}+J\left [1-cos(k)\right ] }

case of 2 flipped spin state

Eigenstate is:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\Psi\rangle = \sum^N_{n1<n2}a(n_1,n_2)|n_1,n_2\rangle }

Bethe's approach for the coefficients Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a(n_1,n_2)} are again plane wave but with unknown amplitudes Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_1} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_2} :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a(n_1,n_2)=A_1e^{i(k_1n_1+k_2n_2)}+A_2e^{i(k_1n_2+k_2n_1)} }

The parameters Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_1 } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_2 } are determined by the insertion into the Schrodinger equation. This gives the following amplitude ratio:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{A_1}{A_2}=e^{i \theta}=-\frac{e^{i(k_1+k_2)}+1-2e^{ik_1}}{e^{i(k_1+k_2)}+1-2e^{ik_2}} }

with:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a(n_1,n_2)=e^{i(k_1n_1+k_2n_2+\frac{1}{2}\theta_{12})}+e^{i(k_1n_2+k_2n_1+\frac{1}{2}\theta_{21})} }

Using periodic boundary conditions we obtain the wave numbers Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_1, k_2 } and the angle Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta = \theta_ {12} =- \theta_ {2,1} } satisfying the following equations:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2\cot \frac{\theta}{2}=\cot\frac{k_1}{2}-\cot\frac{k_2}{2} \qquad Nk_1=2\pi\lambda_1+\theta\qquad Nk_2=2\pi\lambda_2-\theta }

where the integers Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda_i = {0,1 ... N} } are called Bethe quantum numbers. Thus all eigenvectors for 2 flipped spin case is determined by all possible pairs that satisfy the equations. The energy is then given by:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E=E_{0}+J\sum_{j=1,2}\left (1-\cos(k_j)\right ) }

Case of r flipped spin

Eigenstates:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\Psi\rangle = \sum^N_{n1<n2<..<n_r}a(n_1,n_2,..,n_r)|n_1,n_2,..,n_r\rangle }

with coefficients:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a(n_1,..n_r)=\sum_{P\in S_r}\exp\left(i\sum^r_{j=1}k_{P_j}n_j+i\sum_{i<j}\theta_{P_iP_j} \right) }

The sum runs over all possible Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r! } permutation of the numbers Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {1, .., r} } . Inserting into the Schrödinger equation and applying the periodic boundary conditions lead to:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{alignat} \cdot 2 \cot \frac{\theta_{ij}}{2}&=\cot\frac{k_i}{2}-\cot\frac{k_j}{2} &\qquad \text{where}\quad& i,j=1..r \\ Nk_i&=2\pi\lambda_i+\sum_{j \neq i}\theta_{ij}&&\lambda_i={1,..,N-1} \end{alignat} }

The eigenvectors are given with all combinations of Bethe quantum numbers Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\lambda_1, .. \lambda_r) } satisfying the Bethe equations. The energy of the corresponding state is given by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E=E_{0}+J\sum^r_{j=1}\left (1-\cos k_j \right) }